证明有理数乘无理数仍然是无理数.如题.
问题描述：
证明有理数乘无理数仍然是无理数.
如题.
答
可利用反证法,要用到有理数和无理数的定义.
整数和分数统称有理数,也就是说对一个有理数必可表为a/b其中a、b是某个整数,反之不能这样表示的就是无理数.
Proof:Assume x is a rational number and y is a irrational number,
then there exist two integers a,b that x=a/b.
The product of x,y is z=xy=ay/b.(1)
If z is a rational number,then there exist two integer c,d that z=c/d(2)
from(1)(2) we get ay/b=c/d ,that is y=bc/ad.
As we know,a,b,c,d are all integers ,which make y must be a rational number,that is a contravention.
Thus,z must be a irrational number.